https://file.aiquickdraw.com/mathbot/file/1727775457510t4kpdkic.jpg

Let's analyze the equations provided in your image and work through each one to identify the ellipses. We will then determine how to label the parts and coordinates.

### 1. \( \frac{x^2}{16} + \frac{y^2}{12} = 1 \)
This is the equation of an ellipse in standard form, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where:
- \( a^2 = 16 \), so \( a = 4 \)
- \( b^2 = 12 \), so \( b = 2\sqrt{3} \)

This means:
- The center is at \( (0, 0) \).
- The major axis is along the x-axis with a length of \( 2a = 8 \).
- The minor axis is along the y-axis with a length of \( 2b = 4\sqrt{3} \).

### 2. \( x^2 + 4y^2 = 16 \)
We can rewrite this as \( \frac{x^2}{16} + \frac{y^2}{4} = 1 \), which is also the equation of an ellipse where:
- \( a^2 = 16 \), so \( a = 4 \)
- \( b^2 = 4 \), so \( b = 2 \)

This means:
- The center is at \( (0, 0) \).
- The major axis is along the x-axis with a length of \( 2a = 8 \).
- The minor axis is along the y-axis with a length of \( 2b = 4 \).

### 3. \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \)
This is the equation of an ellipse where:
- \( a^2 = 9 \), so \( a = 3 \)
- \( b^2 = 25 \), so \( b = 5 \)

This means:
- The center is at \( (0, 0) \).
- The major axis is along the y-axis (since \( b > a \)) with a length of \( 2b = 10 \).
- The minor axis is along the x-axis with a length of \( 2a = 6 \).

### 4. \( 3x^2 + 2y^2 = 6 \)
Dividing through by 6, we get \( \frac{x^2}{2} + \frac{y^2}{3} = 1 \), which is the equation of an ellipse where:
- \( a^2 = 2 \), so \( a = \sqrt{2} \)
- \( b^2 = 3 \), so \( b = \sqrt{3} \)

This means:
- The center is at \( (0, 0) \).
- The major axis is along the y-axis (since \( b > a \)) with a length of \( 2b = 2\sqrt{3} \).
- The minor axis is along the x-axis with a length of \( 2a = 2\sqrt{2} \).

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Now, would you like further details or help with plotting these ellipses on graphing paper?

Here are 5 related questions:
1. What is the general form of the equation of an ellipse?
2. How do you find the foci of an ellipse?
3. Can an ellipse's axes be rotated, and how would that affect the equation?
4. What are the properties of hyperbolas, and how are they different from ellipses?
5. How can you calculate the eccentricity of an ellipse?

**Tip:** Always double-check your ellipse parameters before plotting, particularly the lengths of the axes, as they directly determine the shape of the graph.

Draw the ellipse and label all parts and coordinates for the following equations:
1. (x^2/16) + (y^2/12) = 1
2. x^2 + 4y^2 = 16
3. (x^2/9) + (y^2/25) = 1
4. 3x^2 + 2y^2 = 6

This tutorial walks through solving and graphing ellipses from equations like (x^2/16) + (y^2/12) = 1. Learn about major and minor axes, calculating a and b values, and the general form of ellipses. Perfect for high school students studying conic sections.

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